Question

Let \({\left( { - 2 - \frac{1}{3}i} \right)^3} = \frac{{x + iy}}{{27}}\left( {i = \sqrt { - 1} } \right)\), where x and y are real numbers, then y – x equals:
1. 91
2. -85
3. 85
4. -91

Answer 1

Correct Answer - Option 1 : 91

From question, the complex number given is:

\(\frac{{x + iy}}{{27}} = {\left( { - 2 - \frac{1}{3}i} \right)^3}\) 

Now,

\(\Rightarrow \frac{{x + iy}}{{27}} = {\left[ {\frac{{ - 1}}{3}\left( {6 + i} \right)} \right]^3}\) 

∵ [(a + b)³ = a³ + b³ + 3a²b + 3ab² and i² = -1]

\(\Rightarrow \frac{{x + iy}}{{27}} = - \frac{1}{{27}}\left( {216 + 108i + 18{i^2} + {i^3}} \right)\)

\(\therefore \frac{{x + iy}}{{27}} = - \frac{1}{{27}}\left( {198 + 107i} \right)\) 

On equating real and imaginary part,

∴ x = -198 and y = -107

From question,

⇒ y – x = -107 + 198

∴ y – x = 91